Hi Nupe, Yep I have lots of air tools including stapler this works fine for hardware cloths.
Mims, that is indeed a very good method.
For most of heavy duty hardware cloth fastening I use screws but I use 1x2 or 1x4 material in place of those washers, by using the boards, there is no part of the hardware cloth that is free or could be pulled loose by any critters, and my wife likes the more finished look (happy wife, happy life). The boards also mean I don't fumble with washers (I would drop quite a few).
I also use 1/4" hardware cloth and for that I use my air stapler and use batten boards the same way I use the 1 by material listed above. Our coop has this material covering all the air vents and on top of the hardware cloth (to the outside) I have metal window screen material held with staples and batten boards. For any predator to get inside they have to go through three layers of increasingly heavier materials and this is after they have managed to get into the run in the first place. By that time I would be there with one of my dispatching tools (rifle or pistol) and it would be the end of their story.
I use a fastener similar to the bigheel models for our 14 and 12 gauge 2" x 4" x 4' tall perimeter fences where they are attached to wood fence poles, the T- poles are where the fencing is wired tight, all our fencing is stretched tight with a come-along before being fastened in place. The hardware cloths are also stretched but they can not be pulled quite as tight as the true fencing.